Design and analysis of centrifugal compressor in carbon dioxide heat pump system

Based on the advantages of energy saving, environmental protection and high efficiency, carbon dioxide heat pump system has great application prospects. However, there are still many technical problems to be solved, especially the design and optimization of carbon dioxide centrifugal compressor. In this paper, a centrifugal compressor in carbon dioxide heat pump system is designed. The compressor is directly driven by a high-speed permanent magnet synchronous motor. Two-stage impellers are installed on both sides of the motor, and the bearings are active magnetic bearings. The influences of inlet pressure and temperature on compressor performance are analyzed. In the range of inlet temperature from 35 to 55 °C, with the decrease of inlet temperature, the compressor pressure ratio increases by 12–29.8%, the power increases by 2.7–8.6%. In the range of inlet pressure from 4 to 6 MPa, with the increase of inlet pressure, the compressor pressure ratio increases by 12.3–38.6%, and the power increases by 8.7–17.8%. In addition, the calculation method of compressor axial force is introduced, the axial force is calculated, analyzed and optimized. Furthermore, the rotor dynamics of compressor rotor and the influences of bearing stiffness and diameter of motor rotor on rotor dynamics are studied. With the increase of bearing stiffness, the first-order critical speed and maximum displacement of the rotor increase. The research provides a theoretical reference for the design and optimization of centrifugal compressor in carbon dioxide heat pump system.


List of symbols P 0
Inlet total pressure of compressor (MPa) p 0 Pressure in front of the impeller inlet hub (MPa) p 1 Inlet pressure of impeller (MPa) p 2 Outlet pressure of impeller (MPa) n Impeller rotating speed (r/min) F 1 Axial force acting on the inlet face of impeller (kN) F 2 Axial force acting on the inlet-outlet part of impeller (kN) F 1m Axial force generated by the change of momentum from axial direction to radial direction (kN) r 1 Radius of impeller inlet (m) d m Diameter of the sealing diameter between the back of the impeller and the motor (m) D 1 Diameter of impeller inlet (m) K xx /K yy Stiffness of the journal bearing (N/mm) F tot1 Axial force of first-stage impeller (kN) d m1 Diameter of the sealing diameter between the back of the first-stage impeller and the motor (m) d r Diameter of motor rotor (mm) T 0 Inlet total temperature of compressor (°C) F tot Axial force of compressor (kN) C 01 Inlet velocity (m/s) Q m Mass Flow Rate (kg/s) D 0 Diameter of impeller inlet hub (m) F 0 Axial force acting on the inlet hub of impeller (kN) F 3 Axial force acting on the impeller back (kN)

Rotor layout
The layout of the carbon dioxide compressor is shown in Fig. 2, and the descriptions of each component are as follows.
( The carbon dioxide centrifugal compressor will be designed according to API 617 34 .Nuts are used to fix the rotor in the axial direction, and the impeller and the motor cavity are sealed by a sealing structure.The motor adopts a permanent magnet synchronous motor, and the compressor impeller is installed on both sides of the motor shaft and directly driven by the motor.Comb seal or dry gas seal can be selected as the sealing structure according to the design requirements.Both the radial bearing and the thrust bearing adopt active magnetic suspension bearings, which have long service life and do not need a lubricating oil system.Most axial force can be offset by the back-to-back design of the first-stage impeller and the second-stage impeller, and the balance of residual axial force in the whole rotor system is finally realized by the thrust bearing.

Compressor sizing
According to the conditions of carbon dioxide heat pump cycle, the operating conditions of the compressor are determined as shown in Table 1.The aerodynamic design of compressor is completed using Compal software 35 .The first-stage design pressure ratio is 1.6, and the second-stage design pressure ratio is 1.43.The parameters of the two-stage impellers are shown in Table 2.The three-phase distribution diagram of carbon dioxide is shown in Fig. 3, and the Inlet state of compressor (point 1) and outlet state (point 2) are identified in Fig. 3. Carbon dioxide at the inlet of the compressor is in a gaseous state, and the carbon dioxide at the outlet after being compressed by the impellers is in supercritical state, so the whole compression process is transcritical.
The layout of the two-stage centrifugation compressor design is shown in Fig. 4. A diffuser and a volute are designed behind each stage impeller, and the outlet of the first stage volute is connected with the inlet of the second stage impeller through an external pipeline.
The performance of compressor can be accurately predicted by Compal software.The performance curves of the compressor are shown in Fig. 5.With the increase of flow rate, the pressure ratio of compressor decreases, the power increases, and the efficiency first increases and then decreases.With the increase of rotating speed, the pressure ratio and power of the compressor gradually increase.At low speed, the efficiency of compressor changes more sharply with the flow rate.The main reason is that the working point deviates greatly from the rated design point when the compressor is running at low speed.
Because carbon dioxide is gaseous at the compressor inlet, as shown in Fig. 6, the density of carbon dioxide is greatly influenced by temperature and pressure.When the temperature is 45 °C and the pressure increases from 4 to 6 MPa, the density of carbon dioxide increases by 74.4%.The density of carbon dioxide increased by 20.5% when the pressure was 4.95 MPa and the temperature was increased from 30 to 60 °C.In the carbon dioxide heat pump cycle, the inlet parameters will also change with the external parameters when the compressor is running, so it is of great significance to study the influence of compressor inlet parameters on compressor performance for compressor control in the cycle.The predicted performance curves of the compressor at 26000 rpm, inlet pressure of 4.95 MPa and inlet temperature from 35 to 55 °C are shown in Fig. 7.In the range of inlet temperature from 35 to 55 °C, with the decrease of inlet temperature, the compressor pressure ratio increases by 12-29.8%, the power increases by 2.7-8.6%, and the efficiency decreases more slowly after the design point.This is mainly because, at the same inlet pressure, with the decrease of temperature, the density of carbon dioxide increases, and the volume flow at the inlet of the compressor decreases, so that a larger pressure ratio can be achieved at the same speed, and similarly, the compressor power will also increase.
The predicted performance curves of the compressor at 26000 rpm, inlet temperature of 45 °C and inlet pressure from 4 to 6 MPa are shown in Fig. 8.In the range of inlet pressure from 4 to 6 MPa, with the increase of inlet pressure, the compressor pressure ratio increases by 12.3-38.6%,and the power increases by 8.7-17.8%.The maximum inlet mass flow of compressor at inlet pressure of 4 MPa is 21.4% lower than that at inlet pressure of 6 MPa.With the increase of pressure, the density of carbon dioxide increases, the volume flow at the inlet of compressor decreases and the pressure ratio increases.When the inlet pressure of the compressor is 6 MPa, the pressure ratio of the compressor changes relatively little with the flow rate, mainly because the higher the inlet pressure, the greater the carbon dioxide density, and the smaller the change of the inlet volume flow rate in the same mass flow range.

Axial force analysis
The centrifugal compressor in carbon dioxide heat pump cycle has high pressure and high density, which leads to a very large axial force of a single impeller and requires high design requirements for axial thrust bearings.Therefore, it is necessary to accurately predict and optimize the axial force of the compressor to ensure its safe and stable operation.The axial force of carbon dioxide compressor is related to many parameters, such as flow rate, sealing diameter, impeller size, impeller inlet and outlet pressure, temperature and so on.In the design process of carbon dioxide compressor, the calculation method of axial force of compressor is as follows.
As shown in Fig. 2, the two-stage impellers of the compressor designed in this study are set back to back, and the axial force directions of the two impellers are opposite, and the axial force of the compressor is as shown in Eq. ( 1).As shown in Fig. 9, the axial force of the single-stage impeller of the compressor is mainly divided into four parts.The gas force F 0 acts on the inlet hub of the compressor impeller, the gas force F 1 acts on the inlet surface, the gas force F 2 acts on the inlet-outlet part of the impeller, and the gas force F 3 acts on the back part of the impeller from the sealing diameter to the outer diameter of the impeller.The relationship between the parts is as follows.
(1) where F tot is the axial force of compressor, F tot1 is the axial force of first-stage impeller, F tot2 is the axial force of second-stage impeller, F tot1,2 is the axial force of the first-stage impeller or the second-stage impeller, D 0 is the diameter of the impeller inlet hub, and p 0 is the pressure in front of the impeller inlet hub.Where F 10 is the axial force generated by the static pressure of inlet gas, and F 1m is the axial force generated by the change of momentum from axial direction to radial direction 36 .The calculation formulas are as follows:   where D 0 is the diameter of impeller inlet hub, D 1 is the diameter of impeller inlet, p 1 is the inlet pressure of impeller, Q m is the inlet mass flow rate, c 01 is the inlet velocity, Suppose that the change rule of p is as follows.
where p 2 is the outlet pressure of impeller, r 1 is the radius of impeller inlet, r 2 is the radius of impeller outlet.
The following relation is obtained: where D 2 is the diameter of impeller outlet.The axial thrust formula on the back of the impeller from the sealing diameter to the outer diameter of the impeller can be expressed as follows: where d m is the diameter of the sealing diameter between the back of the impeller and the motor.
The related dimension parameters of axial force are shown in Table 3.
As shown in Fig. 10, the axial force of the first-stage impeller with different sealing diameters d m1 .With the increase of rotating speed, the axial force direction of the first-stage impeller changes from the reverse direction (inlet flow direction) to the positive direction.Due to the increase of rotating speed, the outlet pressure of the impeller increases and the gas force F 3 in Eq. ( 2) increases.With the increase of mass flow rate, the reverse value of axial force of the first-stage impeller increases and the positive value decreases, mainly because the gas force F 1 in Eq. ( 2) increases and the axial force value of the impeller decreases with the increase of mass flow rate.In the range of dm1 from 30 to 60 mm, with the increase of the diameter of the sealing diameter, the positive value of the axial force of the impeller decreases, the reverse value increases and the variation range of the axial force (5) decreases.In the range of rotating speed from 13000 to 26000 rpm, the axial force varies from − 1.16kN to 8.51kN when sealing diameter d m1 is 30 mm, and from − 12.14kN to − 4.79kN when sealing diameter d m1 is 60 mm.As shown in Fig. 11, the axial force of the second-stage impeller with different sealing diameters d m2 .The axial force variation law of the second-stage impeller is basically the same as that of the first-stage impeller.In the range of rotating speed from 13000 to 26000 rpm, when the sealing diameter d m2 is 30 mm, the axial force of the second-stage impeller changes from − 2.36kN to 9.09kN, and when the sealing diameter d m2 is 60 mm, the axial force changes from − 14.19kN to − 11.63kN.When the sealing diameter d m2 is 60 mm, the axial force of the second-stage impeller has a smaller range than that of the first-stage impeller.
In the range of sealing diameter d m from 30 to 60 mm, the axial force of compressor is shown in Fig. 12.When the first-stage impeller sealing diameter d m1 is constant, the positive value of the axial force of the compressor increases and the negative value decreases when the second-stage impeller sealing diameter d m2 ranges from 30 to 60 mm.When the sealing diameter of the second-stage impeller d m2 is constant, the positive value of the axial force of the compressor decreases and the negative value increases when the sealing diameter of the first-stage impeller d m1 ranges from 30 to 60 mm.Therefore, it is necessary to find two suitable sealing diameters d m1 and d m2 to keep the axial force of the compressor in a suitable fluctuation range and reduce the design difficulty of the thrust bearing.
Combined with the motor rotor size, and through the optimization of the first-stage seal diameter d m1 and the second-stage seal diameter d m2 , it is found that when d m1 = 53.6 mm and d m2 = 47.2 mm, the axial force of the compressor is within ± 1.5kN, and the absolute value and variation range of the axial force are the best.The relationship between the axial force and the flow rate and speed is shown in Fig. 13.
A preliminary three-dimensional model of the compressor is shown in Fig. 14.

Rotor dynamics analysis
The rotor dynamics analysis of compressor rotor is completed by Dyrobes software.The rotor dynamics model is shown in Fig. 15.Shaft elements were used to model the rotor length, and diameter values of nuts, impellers, seal structure, thrust collar and motor rotor.Impellers can be modeled as disks.The mass, diametral moment of inertia and polar moment of inertia of the first-stage impeller are 1.01 kg, 7.198 × 10 −4 m 4 and 1.328 × 10 −3 m 4 respectively, and the mass, diametral moment of inertia and polar moment of inertia of the second-stage impeller are 0.88 kg, 6.095 × 10 −4 m 4 and 1.139 × 10 −3 m 4 respectively.Both radial bearing and thrust bearing adopt active magnetic bearing, and the initial values of stiffness and damping of radial bearing are set to 1000N/mm and 1N•s/mm respectively.The critical speed map of the rotor is shown in Fig. 16.Due to the limitation of the stiffness of magnetic bearing, the critical speed will definitely appear in the operating speed range of carbon dioxide compressor in this study.Therefore, it is necessary to analyze the steady synchronous response of the rotor and further judge whether the compressor rotor is a rigid rotor according to the vibration mode of the rotor.
The goal of the rotor dynamics analysis was to study the forced unbalance response of rotor and evaluate the stability of the rotor 37 .The rotor synchronous response at the first-order critical speed within the operating speed is shown in Fig. 17 and the results show that the vibration mode of the rotor is oscillating vibration mode and belongs to rigid vibration mode.Therefore, according to API Standard 617, the rotor is a rigid rotor.
When the radial bearing stiffness ranges from 1000N/mm to 7000N/mm, the lateral steady state response curves of rotor stations 1, 6, 16, 25 and 31 are shown in Fig. 18.With the increase of bearing stiffness, the firstorder critical speed and maximum displacement of the rotor increase, and the amplification factor (AF) of each station also increases.And the results show that the first-order critical speed of the rotor is lower than the design speed.This is mainly because the stiffness adjustment range of magnetic bearing is limited (generally lower than 1 × 10 4 N/mm).As shown in Fig. 16, the first-order critical speed of the rotor cannot be greater than the design speed by adjusting the stiffness of magnetic bearing.
The lateral steady-state response curves of rotor stations 1, 6, 16, 25 and 31 with different diameters of motor rotor are shown in Fig. 19.When that diameter of the motor rotor is in the range of 60 mm to 120 mm, the maximum displacement of the rotor occurs at the position of Station 1.According to the rotor synchronous response in Fig. 17, the position of Station 1 is the maximum displacement under this vibration mode.With the increase of motor rotor diameter, the maximum displacement of the rotor increases.According to API Standard 617, with the increase of rotor diameter and rotor weight, the allowable unbalance value of the rotor increases, resulting in the increase of the maximum displacement of the rotor.When the motor rotor diameter is 120 mm,  www.nature.com/scientificreports/ the maximum displacement of the compressor rotor is 176.6% higher than that when the motor rotor diameter is 60 mm.

Results and discussion
In this paper, a centrifugal compressor in carbon dioxide heat pump system is designed.The compressor is directly driven by a high-speed permanent magnet synchronous motor.Two-stage impellers are installed on both sides of the motor, and the bearings are active magnetic bearings.The influences of inlet pressure and temperature on compressor performance are analyzed, the calculation method of compressor axial force is introduced, the axial force is calculated, analyzed and optimized, the rotor dynamics of compressor rotor is analyzed, and the influence of bearing stiffness on rotor dynamics is studied.The conclusions are summarized as follows.
(1) The density of carbon dioxide is greatly influenced by temperature and pressure.When the temperature is 45 °C and the pressure increases from 4 to 6 MPa, the density of carbon dioxide increases by 74.4%.The density of carbon dioxide increased by 20.5% when the pressure was 4.95 MPa and the temperature was increased from 30 to 60 °C.
In the range of inlet temperature from 35 to 55 °C, with the decrease of inlet temperature, the compressor pressure ratio increases by 12-29.8%, the power increases by 2.7-8.6%, and the efficiency decreases more slowly after the design point.In the range of inlet pressure from 4 to 6 MPa, with the increase of inlet pressure, the   (2) With the increase of rotating speed, the axial force direction of the first-stage impeller changes from the reverse direction (inlet flow direction) to the positive direction.the axial force varies from − 1.16kN to 8.51kN when sealing diameter d m1 is 30 mm, and from − 12.14kN to − 4.79kN when sealing diameter d m1 is 60 mm.When the first-stage impeller sealing diameter d m1 is constant, the positive value of the axial force of the compressor increases and the negative value decreases when the second-stage impeller sealing diameter d m2 ranges from 30 to 60 mm.Through the optimization of the first-stage seal diameter d m1 and the second-stage seal diameter d m2 , it is found that when d m1 = 53.6 mm and d m2 = 47.2 mm, the absolute value and variation range of the axial force of the compressor are the best.
(3) The vibration mode of the rotor is oscillating vibration mode and the rotor is a rigid rotor.With the increase of bearing stiffness, the first-order critical speed and maximum displacement of the rotor increase, and the amplification factor (AF) of each station also increases.When the motor rotor diameter is 120 mm, the maximum displacement of the compressor rotor is 176.6% higher than that when the motor rotor diameter is 60 mm.
The analysis results in this paper can provide reference for the design of centrifugal compressor in carbon dioxide heat pump system.In the future research work, it is necessary to test and verify the design and analysis results and increase the design power of carbon dioxide heat pump system and centrifugal compressor to ensure their wider popularization and use.

Figure 4 .
Figure 4. Layout of the Two-Stage Compressor Design.
(a) Pressure ratio total-total versus Q m (b) Power versus Q m (c) Adiabatic efficiency total-total versus Q

Figure 6 .
Figure 6.Variation diagram of carbon dioxide density with temperature and pressure.

Figure 7 .
Figure 7. Performance curves of the compressor at 26000 rpm with different inlet temperatures.

Figure 8 .
Figure 8. Performance curves of the compressor at 26000 rpm with different inlet pressures.

Figure 14 .
Figure 14.Preliminary three-dimensional model of the compressor.

Figure 15 .
Figure 15.Rotor dynamics model of compressor. A.

Table 1 .
Operating conditions for compressor design.
Figure 3. Carbon dioxide three-phase distribution diagram.